Furthermore, we demonstrate the execution speed of each algorithm across different graph sizes. Compared to the original GOT formulation presented in [1]}, we can observe a significant speed improvement in both of our fGOT algorithms. In particular, using the corresponding filter \(g(L) = \sqrt{L^{\dagger }} \) , the MGD algorithm renders the computation of the original GOT distance comparable to the GW distance speed, circumventing the largest drawback of the GOT framework.
The stochastic MGD algorithm also offers a great speed improvement, while successfully accounting for the nonconvexity of the alignment problem. This allows the algorithm to capitalise on both speed and accuracy (in terms of lower \(l_2\)  distance), offering better performance than its non-stochastic counterpart at a faster rate than GOT.
